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Transformations of Functions
Transformations alter a function while maintaining the original characteristics of that funcction.
Learning Objective

Differentiate between three common types of translations: reflections, rotations, and scaling
Key Points
 Transformations are ways that a function can be adjusted to create new functions.
 Transformations often preserve the original shape of the function.
 Common types of transformations include rotations, translations, reflections, and scaling (also known as stretching/shrinking).
Terms

translation
Shift of an entire function in a specific direction

Scaling
changes the size and/or the shape of the function

rotation
Spins the function around the origin

reflection
Mirror image of a function
Full Text
A transformation takes a basic function and changes it slightly with predetermined methods. This change will cause the graph of the function to move, shift, or stretch, depending on the type of transformation. We will talk about four main types of transformations: translations, reflections, rotations, and scaling.
Translations
A translation moves every point by a fixed distance in the same direction. The movement is caused by the addition or subtraction of a constant from a function. As an example, let
Reflections
A reflection of a function causes the graph to appear as a mirror image of the original function. This is caused by switching the sign of the input going into the function. Let the function in question be
Rotations
A rotation is a transformation that is performed by "spinning" the object around a fixed point known as the center of rotation. Although the concept is simple, it has the most advanced mathematical process of the transformations discussed. There are two formulas that are used:
Scaling
Scaling is a transformation that changes the size and/or the shape of the graph of the function. Note that until now, none of the transformations we discussed could change the size and shape of a function  they only moved the graphical output from one set of points to another set of points. As an example, let
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Key Term Reference
 constant
 Appears in these related concepts: Graphing Quadratic Equations in Vertex Form, Inverse Variation, and Direct Variation
 degree
 Appears in these related concepts: Solving Quadratic Equations by Factoring, Partial Fractions, and Adding and Subtracting Polynomials
 distance
 Appears in these related concepts: Inequalities with Absolute Value, The Distance Formula and Midpoints of Segments, and Linear Mathematical Models
 expression
 Appears in these related concepts: Compound Inequalities, Sets of Numbers, and Simplifying, Multiplying, and Dividing
 function
 Appears in these related concepts: Visualizing Domain and Range, The Vertical Line Test, and Solving Differential Equations
 graph
 Appears in these related concepts: Graphical Representations of Functions, Graphing Equations, and Graphs of Equations as Graphs of Solutions
 output
 Appears in these related concepts: Functions and Their Notation, A Study of Process, and Introducing Aggregate Supply
 point
 Appears in these related concepts: Graphing Quadratic Equations In Standard Form, Relative Minima and Maxima, and Polynomial and Rational Functions as Models
 set
 Appears in these related concepts: Sequences, Introduction to Sequences, and Sequences of Mathematical Statements
 sign
 Appears in these related concepts: Polynomial Inequalities, The Intermediate Value Theorem, and The Rule of Signs
 transformation
 Appears in these related concepts: Horizontal Gene Transfer, Prokaryotic Reproduction, and Reflections
Sources
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Cite This Source
Source: Boundless. “Transformations of Functions.” Boundless Algebra. Boundless, 17 Jun. 2016. Retrieved 28 Sep. 2016 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/functions4/transformations31/transformationsoffunctions1175543/